Get all pairs of same consecutive characters from string?

str = "AabHHioPjggtYuggbwq";
StringCases[str, x_ ~~ x_, IgnoreCase -> True]

{"Aa", "HH", "gg", "gg"}

Or in the generalized version of the question:

str = "AaAbHHhhhioPjggtYuggGgGbwq";
StringCases[str, x_ ~~ (x_) .., IgnoreCase -> True]

{"AaA", "HHhhh", "gg", "ggGgG"}

getpairs[str_String] := StringJoin @@@ Select[
   Partition[Characters[str], 2, 1],
   SameQ @@ ToUpperCase[#] &]

str = "AabHHioPjggtYuggbwq";

getpairs[str]

(*  {"Aa", "HH", "gg", "gg"}  *)

I will give a RegularExpression version here.

str = "AabHHioPjggtYuggbwq";
StringCases[str, RegularExpression["(?i)(\\w)\\1"]]

{"Aa", "HH", "gg", "gg"}

---

Per this comment,that the string contain 3 or more letter case

str = "aAabHhHioPjggtYuggbwwq";
StringCases[str, RegularExpression["(?i)(\\w)\\1+"]]

{"aAa", "HhH", "gg", "gg", "ww"}